Why a Replicating Forward Portfolio Has Zero Initial Value

Summary

The document discusses a one-period, two-state replication of a forward using stock and cash. The stock holding is chosen to match the difference between the forward payoffs across the two future states, and the cash position makes the combined portfolio reproduce those payoffs. The question asks why the initial value of this replicating portfolio is zero when the strike is set to the stock price grown at the risk-free rate.

The author recognizes that the forward itself has zero initial value and finds the stated strike by setting the portfolio value to zero, but the text contains no answer resolving whether or why this step is valid. Under risk-neutral pricing, the portfolio value is the discounted expected payoff, and a zero-cost forward corresponds to choosing the strike so that this value is zero. The document is therefore a useful pricing question, but leaves its central derivation unfinished.

Key ideas

• A stock and cash position can replicate a forward payoff across two future states.
• Risk-neutral valuation expresses the portfolio value as the discounted expected payoff.
• A zero initial forward value requires a strike that makes the discounted expected payoff zero.
• The supplied discussion poses the replication question but does not provide a complete derivation.

Tags

# Is the initial value of the portfolio replicating a forward zero?


# Is the initial value of the portfolio replicating a forward zero?












This is from the book Financial Calculus: An Introduction to Derivative Pricing by Martin Baxter.

By choosing appropriate weights in a portfolio of a stock and cash bond you can replicate the payoff of a forward.

Notation: we are now in State 1, with stock price $S_1$. In the future we will be in State 2 (with stock price $S_2$) with r.n. prob $1-q$, or in State 3 (stock price $S_3$) with r.n. probability $q$. The variable $f$ represents the payoff of the forward ($f_n=S_n-K$).

The value of the portfolio is $$V=s_{1}\frac{f(3)-f(2)}{s_{3}-s_{2}}+e^{-r\delta t}\left(f(3)-\frac{\left(f(3)-f(2)\right)s_{3}}{s_{3}-s_{2}}\right) $$

This all makes sense to me and I understand how under a risk-neutral measure.

$$V = e^{-r \delta t}\left(\left(1-q\right)f(2)+qf(3)\right)$$

In one the exercises, the author asks for us to prove that using $V$, the strike price is $$S_1e^{r \delta t}$$.

I substitute $f(2)=s_{2}-k$ and $f(3)=s_{3}-k$ but I don’t end up with the desired answer. However I’m able to get $K= S_1e^{r \delta t}$ if I let $V=0$.

Is is it correct to let $V=0$? Or have I missed anything here? On one hand I know that initial value of the forward is 0 but I can’t get my head around how the initial value of the portfolio we have set up is also 0.

Thanks in advance!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.